Live Exercise 3: Optimal Patent Length — the Ideas Model

MSc-level Industrial Organisation course at the University of St Andrews
Author

Gerhard Riener

Solutions

Group exercise (≈20 minutes)

  • Work in groups of 2–3.
  • Show all intermediate steps.
  • Parts (a)–(c) are computational; part (d) is a short discussion.

Problem: Patent length and the ideas model

A patent regulator is evaluating two candidate drugs under the Scotchmer ideas model. Each drug is characterised by a pair \((\nu, F)\), where \(\nu\) is the per-period consumer surplus under competitive supply and \(F\) is the fixed development cost.

The regulatory parameters are:

\[ \pi = \tfrac{1}{2}, \qquad \lambda = \tfrac{1}{4}, \qquad r = \tfrac{1}{4}, \]

where \(\pi\) is the share of per-period consumer surplus appropriated by the patent holder as profit, \(\lambda\) is the per-period deadweight loss as a share of \(\nu\), and \(r\) is the discount rate. The current (discounted) patent length is \(T = 20\).

Drug \(\nu\) \(F\)
Alpha 10 60
Beta 5 10

(a) Private investment condition

For each drug, determine whether a firm will voluntarily invest given \(T = 20\).

The investment condition is: \(\pi \nu T \ge F\).

(b) Net social value

For each drug, compute the net discounted social value of development:

\[ \text{Social value} = \frac{\nu}{r} - \lambda \nu T - F. \]

(Note: with \(r = \tfrac{1}{4}\), the perpetual benefit per unit of \(\nu\) is \(\tfrac{1}{r} = 4\).)

Does either drug yield a positive net social surplus at \(T = 20\)?

(c) Socially optimal patent length for Drug Beta

Find the minimum patent length \(T^*\) that just induces private investment in Drug Beta. At \(T = T^*\), compute the net social value and state your conclusion.

(d) Discussion (5 minutes)

At \(T = 20\), both drugs are privately profitable yet socially wasteful. Drug Beta becomes socially efficient at a much shorter patent length.

  1. What does this imply for the design of a uniform patent length (the same \(T\) for all drugs)?
  2. Why is it difficult in practice to implement drug-specific patent lengths, even if they would be welfare-improving?

Solution

Parameters: \(\pi = \tfrac{1}{2}\), \(\lambda = \tfrac{1}{4}\), \(r = \tfrac{1}{4}\) (so \(\tfrac{1}{r} = 4\)), \(T = 20\).

(a) Private investment condition

Drug Alpha (\(\nu = 10\), \(F = 60\)):

\[ \pi\nu T = \tfrac{1}{2}\times 10\times 20 = 100 \ge 60. \qquad \checkmark \quad \text{Invests.} \]

Drug Beta (\(\nu = 5\), \(F = 10\)):

\[ \pi\nu T = \tfrac{1}{2}\times 5\times 20 = 50 \ge 10. \qquad \checkmark \quad \text{Invests.} \]

Both drugs are privately profitable at \(T = 20\).

(b) Net social value

\[ \text{Social value} = \frac{\nu}{r} - \lambda\nu T - F. \]

Drug Alpha:

\[ \frac{10}{0.25} - \frac{1}{4}\times 10\times 20 - 60 = 40 - 50 - 60 = -70 < 0. \qquad \text{Socially wasteful.} \]

Drug Beta:

\[ \frac{5}{0.25} - \frac{1}{4}\times 5\times 20 - 10 = 20 - 25 - 10 = -15 < 0. \qquad \text{Socially wasteful.} \]

Neither drug yields positive net social surplus at \(T = 20\). The discounted deadweight loss during the long patent period (\(\lambda\nu T\)) exceeds the present value of social benefit (\(\nu/r\)), net of development cost.

Intuition: With \(r = 1/4\), the present value of the perpetual benefit stream is only \(\nu/r = 4\nu\) — the high discount rate makes future competitive benefits worth little today. Meanwhile, 20 periods of monopoly pricing generate a large deadweight loss.

(c) Socially optimal patent length for Drug Beta

Set the investment condition to equality and solve for \(T^*\):

\[ \pi\nu T^* = F \;\Rightarrow\; \tfrac{1}{2}\times 5\times T^* = 10 \;\Rightarrow\; T^* = 4. \]

Social value at \(T^* = 4\):

\[ \frac{5}{\,1/4\,} - \frac{1}{4}\times 5\times 4 - 10 = 20 - 5 - 10 = 5 > 0. \qquad \checkmark \]

A patent of length \(T^* = 4\) is sufficient to induce investment in Drug Beta and generates a positive net social surplus of 5. The current 20-period patent is five times longer than necessary: the excess deadweight loss from periods 5–20 is a pure social waste — Drug Beta would have been developed anyway.

(d) Discussion

Uniform patent length: A single \(T\) must simultaneously incentivise high-cost drugs (large \(F\), need long protection) and avoid over-rewarding low-cost drugs (small \(F\), short protection suffices). With heterogeneous \((\nu, F)\) pairs, no uniform \(T\) can be first-best: it will either leave high-\(F\) drugs underdeveloped or generate excessive deadweight loss for low-\(F\) drugs.

Drug-specific patent lengths: A regulator would need to observe each drug’s \((\nu, F)\), which are private information held by the firm. Firms have strong incentives to overstate \(F\) to obtain longer protection. Solving this requires screening mechanisms or revelation procedures that are administratively complex and litigation-prone. In practice, partial adjustments are made through supplementary protection certificates, compulsory licensing provisions, and tiered examination standards — but a clean drug-by-drug \(T^*\) remains infeasible.